Thermodynamics of a qubit undergoing dephasing

(1)

This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 140.105.250.36

This content was downloaded on 14/06/2017 at 10:15 Please note that terms and conditions apply.

Thermodynamics of a qubit undergoing dephasing

View the table of contents for this issue, or go to the journal homepage for more 2017 J. Phys.: Conf. Ser. 841 012019

(http://iopscience.iop.org/1742-6596/841/1/012019)

Thermodynamics of a qubit undergoing dephasing

S Marcantoni1,2

1

Dipartimento di Fisica, Universit`a di Trieste, I-34151 Trieste, Italy 2

Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, I-34151 Trieste, Italy E-mail: stefano.marcantoni@ts.infn.it

Abstract. The thermodynamics of a qubit undergoing dephasing due to the coupling with the external environment is discussed. First of all, we assume the dynamics of the system to be described by a master equation in Lindblad form. In this framework, we review a standard formulation of the first and second law of thermodynamics that has been known in literature for a long time. After that, we explicitly model the environment with a set of quantum harmonic oscillators choosing the interaction such that the global dynamics of system and bath is analytically solvable and the Lindblad master equation is recovered in the weak-coupling limit. In this generalized setting, we can show that the correlations between system and bath play a fundamental role in the heat exchange. Moreover, the internal entropy production of the qubit is proven to be positive for arbitrary coupling strength.

1. Introduction

Standard thermodynamics is a phenomenological theory describing transformations of macroscopic systems in equilibrium according to few simple laws [1]. In particular, the first law of thermodynamics deals with the conservation of total energy that can be exchanged in the form of heat or work. The second law of thermodynamics instead regards the entropy, a state function that can never decrease in isolated systems, thus providing an arrow of time.

In the last years many attempts have been made to study the fate of thermodynamics when the system of interest can no longer be considered macroscopic and is out of equilibrium. This research is motivated by the high degree of control reached in many experimental setups such as ultracold atoms [2], optomechanical systems [3] and trapped ions [4], that allows to test the laws of thermodynamics in a completely new scenario. At the nanoscale quantum effects are expected to become important and the formulation of a consistent theory of quantum thermodynamics has recently attracted lot of attention [5]. From a fundamental point of view, one can address the issue of equilibration and thermalization in closed systems [6]. Moreover, it is interesting to study how the phenomenological laws of thermodynamics emerge from the underlying microscopic dynamics and in what sense they can be generalized beyond the usual thermodynamic limit [7].

(3)

2

the important role of bipartite correlations in the heat balance following the definitions given in [10]. Moreover, we can show that the internal entropy production for the qubit is positive without resorting to the weak-coupling approximation.

2. Lindblad master equation

Consider a qubit evolving in time according to the following master equation in Lindblad form [11, 12] ∂t̺(t) = −ihω0 2 σz, ̺(t) i +γ 2 σz̺(t)σz− ̺(t) , (1)

whose solution is easily found to be

̺(t) = ̺00|0ih0| + ̺11|1ih1| + e−γt(̺10eiω0t|1ih0| + ̺01e−iω0t|0ih1|), (2)

whereσz|ℓi = (−)ℓ|ℓi, with ℓ ∈ {0, 1}. This is a model for dephasing, since the populations remain

constant while coherence decays exponentially in time. The thermodynamics of this quantum system can be studied using the standard formulation presented in [8]. In particular, the first law of thermodynamics for a driven open quantum system with HamiltonianH(t) reads

∂tU(t) = ∂tQ(t) + ∂tW(t), (3)

where the internal energy U(t), the heat flux ∂tQ(t) and the work power ∂tW(t) are defined respectively

U(t) := Tr [̺(t) H(t)] , (4) ∂tQ(t) := Tr [∂t̺(t) H(t)] , (5)

∂tW(t) := Tr [̺(t) ∂tH(t)] . (6)

This separation between work and heat contributions to the energy variation is quite reasonable. Indeed, the work power (Eq. (6)) is vanishing as expected in absence of an external field modelled by a time-dependent Hamiltonian, while the heat flux (Eq. (5)) is zero for the unitary dynamics generated byH(t), namely if the system is closed.

For the qubit described in equation (2), the preservation of populations implies that the internal energy, namely the mean value ofσzω0/2, is constant in time. Moreover, the work power is zero, since there is

no time-dependence in the Hamiltonian, and as a consequence the heat exchange with the environment is also vanishing. Up to now, it seems that dephasing happens without energy transfer; however, we will come back to this point later on, when we explicitly consider the presence of the bath.

For an open quantum system interacting with a thermal bath at inverse temperature β, the internal entropy productionσ can be defined as the difference between the total variation of the entropy and the entropy flux due to the heat exchange [13]

σ(t) := ∂tS(t) − β ∂tQ(t), (7)

where S is the von Neumann entropy

S_{(t) := −Tr [̺(t) log ̺(t)] .} (8) The second law of thermodynamics states that_{σ ≥ 0. In our model there is no heat exchange, so that the} internal entropy production equals the variation of the von Neumann entropy and the second law reads ∂tS(t) ≥ 0. Such a property is easily verified using the eigenvalues 1 ± r(t)/2 of the density matrix

(4)

3. Explicit model for the environment

Instead of considering the thermodynamics of the qubit alone, a complete thermodynamic description of the qubit and the environment together could be given, in order to include possible effects due to the correlations between them. Indeed, as shown in [10], in a generic bipartite quantum system the correlations between the subsystems play a fundamental role and a consistent formulation of the first and second law of thermodynamics explicitly accounting for that can be given.

In the following, we study the exactly solvable model of a qubit in interaction with a thermal bosonic bath presented in [9]. In the weak-coupling limit the dynamics of the qubit is given by equation (1), but this more general approach highlights some interesting thermodynamic features. For instance, we show that due to correlations a non zero heat flux in the bath can happen without changing the internal energy of the qubit. Moreover, the internal entropy production of the qubit is found to be positive for an arbitrary coupling strengthλ, namely without performing the standard Markov approximations.

3.1. The model

Consider a total Hamiltonian given byHtot= HS+ HB+ Hintwith

HS = ω0 2 σz, HB= ∞ X k=1 ωka†kak, Hint= λσz⊗ ∞ X k=1 f_k∗ak+ fka†k ,

whereakis the bosonic annihilation operator of modek, satisfying the canonical commutation relations

[ak, a†_l] = δkl, and the complex parametersfkare such thatP∞_k=1|fk|2< ∞. We assume that the initial

state of the total system can be written as̺SB(0) = ̺S(0) ⊗ ̺β_B, where̺S(0) is the initial state of the

qubit and̺β_Bis the Gibbs state of the thermal bath at inverse temperatureβ,

̺S(0) = 1

X

ℓ,ℓ′₌₀

̺ℓℓ′|ℓihℓ′| , σ_z|ℓi = (−)ℓ|ℓi, ̺β

B =

e−βPkωka†kak

Trhe−βPkωka†kak

i. (11)

The dynamics of the total system can be analytically solved (see [9] or [10] for all the details) and it turns out that the density matrix̺SB(t) can be written as follows

̺SB(t) = 1 X ℓ,ℓ′₌₀ ̺ℓℓ′e−iω0ζℓℓ′t/2|ℓihℓ′| ⊗ D ℓ(gt) ̺β_BD†_ℓ′(gt), (12) whereζℓℓ′ = (−)ℓ− (−)ℓ ′

andDℓ(gt) is the displacement operator

Dℓ(gt) = e(−) ℓ λPk[gk(t)a†k−g ∗ k(t)ak]_, _g∗ k(t) = fk∗(e−iωkt− 1)/ωk. (13)

By partial tracing one can obtain the reduced density matrices of the two subsystems

̺S(t) =̺00|0ih0| + ̺11|1ih1| + e−8λ

2

Γ(t) _̺

10eiω0t|1ih0| + ̺01e−iω0t|0ih1|, (14)

(5)

4

The bath at any time is described by a convex combination of displaced thermal states, while the dynamics of the qubit is similar to that one in equation (2) but with a time-dependent dampingΓ(t).

It is possible to recover equation (2) from equation (14) in the so called weak-coupling limit. First of all, we should substitute the discrete sum in equation (16) with the following integral

Γ(t) = Z ∞ 0 dω1 ω coth(βω/2) sin 2_{(ωt/2) e}−ǫω_, ǫ ≥ 0 (17) where a regularized Ohmic spectral density given byfk ≃√ω e−ωǫ/2has been used as in [9]. Then, a

new time scaleτ is defined such that t = τ /λ2and in the limit_{λ → 0 one finds λ}2Γ(τ /λ2_{) ≃ πτ/(2β).} The solution of the Lindblad master equation (1) is immediately found identifying the constant damping rateγ = 4π/β.

3.2. Heat balance

Using the model above, we can exploit an interesting feature of bipartite quantum systems related to the heat balance. The total energy of the composite system Utot defined as follows

U_tot := Tr [Htot̺SB(t)] (18)

is conserved, so that its time derivative is vanishing∂tUtot= 0. Explicitly, one can write

∂tUtot =Tr [Htot∂t̺S(t) ⊗ ̺B(t)] + Tr [Htot̺S(t) ⊗ ∂t̺B(t)] + Tr [Htot∂tχ(t)] =

=Tr∂t̺S(t) HS′(t)

+ Tr∂t̺B(t) HB′ (t)

+ Tr[Hint∂tχ(t)] = 0, (19)

where the operator χ(t) := ̺SB(t) − ̺S(t) ⊗ ̺B(t) has been introduced to describe the amount of

correlations betweenS and B, and a modified Hamiltonian has been defined for each subsystem

H_S,B′ (t) := HS,B+ TrB,S[̺B,S(t)Hint] . (20)

The first two terms in the second line of equation (19) can be associated to heat exchanged byS and B respectively, indeed they are formally similar to equation (5). However, the modified Hamiltonians appear instead of the free HamiltoniansHSandHB; this is expected since the interaction Hamiltonian

should contribute to the internal energy of each subsystem. Given the second line of equation (19) and defining a binding energy as

Uχ(t) := Tr[χ(t)Hint], (21)

the heat balance can be stated as follows

∂tQS(t) + ∂tQB(t) = −∂tUχ(t). (22)

The binding energy is interpreted as an amount of energy stored in the correlations betweenS and B that can be exchanged with both subsystems in the form of heat [10].

For the model of interest, denoting the qubit polarization at time _{t = 0 by hσ}zi and using

equations (14) and (15), the modified qubit Hamiltonian takes the form

(6)

whileH′ B(t) reads H_B′ (t) =X k ωka†_kak+ λ hσzi X k f_k∗ak+ fka†_k . (25)

As a consequence, the heat fluxes for bothS and B can be explicitly computed and turn out to be

∂tQS(t) = 0, (26)

∂tQB(t) = 4λ2 1 − hσzi2∂t∆(t). (27)

Notice that the heat exchanged by the qubit is vanishing even in this more general situation, because the modified Hamiltonian is proportional toσzandTr [∂t̺S(t)σz] = 0 since populations are preserved (see

Eq. (14)). Nevertheless, the heat transfer for the bath of harmonic oscillators does not vanish as expected, indeed the correlations built in during the time-evolution effectively act as a third subsystem exchanging energy withB in the form of heat according to equation (22).

3.3. Entropy production

Another interesting comparison between the present model and the weak-coupling limit discussed in Section 2 regards the internal entropy production for the qubit. Again, the internal entropy production equals the time derivative of the von Neumann entropy because∂tQS(t) = 0. Using equation (14), the

entropy variation of the qubit∂tSS can be calculated from the eigenvalues 1 ± rS(t)

/2 of ̺S(t) with rS(t) = q 1 − 4 ̺00̺11− e−16λ 2 Γ(t)_|̺ 01|2 . (28)

Explicitly, one finds

∂tSS(t) = −1 2log 1 + rS(t) 1 − rS(t) ∂trS(t) = λ216|̺01| 2_e−16λ2 Γ(t) rS(t) log 1 + rS(t) 1 − rS(t) ∂tΓ(t); (29)

therefore, the sign of∂tSS(t) corresponds to the sign of ∂tΓ(t). The behaviour of the latter can be studied

considering the integral expression in equation (17). By means of the substitution_{ωt = e}ω and taking the derivative inside the integral it turns out that

∂tΓ(t) = Z ∞ 0 de ω1 e ωsin 2_(e_{ω/2) ∂} t h coth(β eω/2t) e−ǫeω/ti_{≥ 0.} (30) The second law of thermodynamics in the form ∂tSS(t) ≥ 0 is hence satisfied without invoking the

weak-coupling limit.

4. Conclusions

(7)

6

References

[1] Callen H B 1985 Thermodynamics and an Introduction to Thermostatistics (New York: John Wiley & Sons). [2] Bloch I, Dalibard J and Zwerger W 2008 Rev. Mod. Phys. 80, 885.

[3] Aspelmeyer M, Kippenberg T J and Marquardt F 2014 Rev. Mod. Phys. 86 1391. [4] Leibfried D, Blatt R, Monroe C and Wineland D 2003 Rev. Mod. Phys. 75 281. [5] Vinjanampathy S and Anders J 2016 Contemp. Phys. 57 1-35.

[6] Gogolin C and Eisert J 2016 Rep. Prog. Phys. 79 056001.

[7] Gemmer J, Michel M and Mahler G 2009 Quantum Thermodynamics (Berlin: Springer). [8] Alicki R 1979 J. Phys. A: Math. Gen. 12 103.

[9] Palma G M, Suominen K A and Ekert A 1996 Proc. R. Soc. Lond. A 452 567.

[10] Alipour S, Benatti F, Bakhshinezhad F, Afsary M, Marcantoni S and Rezakhani A T 2016 Sci. Rep. 6 35568. [11] Lindblad G 1976 Commun. Math. Phys. 48 119.